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The extended hypergeometric class of Lévy processes

Part of:Stochastic processes

Published online by Cambridge University Press:  30 March 2016

A. E. Kyprianou,
J. C. Pardo and
A. R. Watson
A. E. Kyprianou
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK. Email address: a.kyprianou@bath.ac.uk.
J. C. Pardo
Affiliation:
CIMAT, CIMAT, Calle Jalisco s/n, Mineral de Valenciana, 36240 Guanajuato, Mexico. Email address: jcpardo@cimat.mx.
A. R. Watson
Affiliation:
Department of Mathematics, University of Zürich, Winterthurerstrasse 190, 8057 Zürich, Switzerland. Email address: alexander.watson@math.uzh.ch.
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Abstract

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We review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (2013) with a view to computing fluctuation identities related to stable processes. We give the Wiener-Hopf factorisation of a process in the extended class, characterise its exponential functional, and give three concrete examples arising from transformations of stable processes.

Keywords

Lévy processhypergeometric Lévy processextended hypergeometric Lévy processWiener-Hopf factorisationexponential functionalstable processpath-censored stable processconditioned stable processhitting distributionhitting probability

MSC classification

Primary:60G51: Processes with independent increments; L'evy processes60G18: Self-similar processes60G52: Stable processes

Information

Type
Part 8. Markov processes and renewal theory
Information
Journal of Applied Probability , Volume 51 , Issue A: Celebrating 50 Years of The Applied Probability Trust , December 2014, pp. 391 - 408
Copyright
Copyright © Applied Probability Trust 2014 

References

Bertoin, J. (1996). Lévy Processes (Camb. Tracts Math. 121). Cambridge University Press.Google Scholar
Bertoin, J., and Yor, M. (2002). The entrance laws of self-similar Markov processes and exponential functionals of Lévy processes. Potential Anal. 17, 389–400.CrossRefGoogle Scholar
Bertoin, J., and Yor, M. (2005). Exponential functionals of Lévy processes. Prob. Surveys 2, 191–212.CrossRefGoogle Scholar
Blumenthal, R. M., and Getoor, R. K. (1968). Markov Processes and Potential Theory (Pure Appl. Math. 29). Academic Press, New York.Google Scholar
Caballero, M. E., and Chaumont, L. (2006). Conditioned stable Lévy processes and the Lamperti representation. J. Appl. Prob. 43, 967–983.Google Scholar
Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2010). On Lamperti stable processes. Prob. Math. Statist. 30, 1–28.Google Scholar
Caballero, M. E., Pardo, J. C. and Pérez, J. L. (2011). Explicit identities for Lévy processes associated to symmetric stable processes. Bernoulli 17, 34–59.Google Scholar
Chaumont, L., and Pardo, J. C. (2006). The lower envelope of positive self-similar Markov processes. Electron. J. Prob. 11, 1321–1341.CrossRefGoogle Scholar
Chaumont, L., Kyprianou, A. E., and Pardo, J. C. (2009). Some explicit identities associated with positive self-similar Markov processes. Stoch. Process. Appl. 119, 980–1000.CrossRefGoogle Scholar
Chaumont, L., Pantí, H., and Rivero, V. (2013). The Lamperti representation of real-valued self-similar Markov processes. Bernoulli 19, 2494–2523.CrossRefGoogle Scholar
Cordero, F. (2010). On the excursion theory for the symmetric stable lévy processes with index α in 1,2 and some applications. Doctoral Thesis, Université Pierre et Marie Curie - Paris VI.Google Scholar
Doney, R. A. (1987). On Wiener-Hopf factorisation and the distribution of extrema for certain stable processes. Ann. Prob. 15, 1352–1362.Google Scholar
Gnedin, A. V. (2010). Regeneration in random combinatorial structures. Prob. Surveys 7, 105–156.Google Scholar
Gradshteyn, I. S., and Ryzhik, I. M. (2007). Table of Integrals, Series, and Products, 7th edn. Elsevier Academic Press, Amsterdam.Google Scholar
Kuznetsov, A. (2010). Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Ann. Appl. Prob. 20, 1801–1830.Google Scholar
Kuznetsov, A. (2010). Wiener-Hopf factorization for a family of Lévy processes related to theta functions. J. Appl. Prob. 47, 1023–1033.Google Scholar
Kuznetsov, A., and Pardo, J. C. (2013). Fluctuations of stable processes and exponential functionals of hypergeometric Lévy processes. Acta Appl. Math. 123, 113–139.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. E., and Pardo, J. C. (2012). Meromorphic Lévy processes and their fluctuation identities. Ann. Appl. Prob. 22, 1101–1135.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. E., and Rivero, V. (2013). The theory of scale functions for spectrally negative Lévy processes. In Lévy Matters II (Lecture Notes Math. 2061), Springer, Heidelberg, pp. 97–186.Google Scholar
Kuznetsov, A., Kyprianou, A. E., Pardo, J. C., and van Schaik, K. (2011). A Wiener-Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Prob. 21, 2171–2190.CrossRefGoogle Scholar
Kuznetsov, A., Kyprianou, A. E., Pardo, J. C., and Watson, A. R. (2014). The hitting time of zero for a stable process. Electron. J. Prob. 19, 1–26.CrossRefGoogle Scholar
Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Kyprianou, A. E., and Patie, P. (2011). A Ciesielski-Taylor type identity for positive self-similar Markov processes. Ann. Inst. H. Poincaré Prob. Statist. 47, 917–928.CrossRefGoogle Scholar
Kyprianou, A. E., and Rivero, V. (2008). Special, conjugate and complete scale functions for spectrally negative Lévy processes. Electron. J. Prob. 13, 1672–1701.Google Scholar
Kyprianou, A. E., Pardo, J. C., and Rivero, V. (2010). Exact and asymptotic n#-tuple laws at first and last passage. Ann. Appl. Prob. 20, 522–564.Google Scholar
Kyprianou, A. E., Pardo, J. C., and Watson, A. R. (2014). Hitting distributions of {α#-stable processes via path censoring and self-similarity. Ann. Prob. 42, 398–430.CrossRefGoogle Scholar
Lamperti, J. (1972). Semi-stable Markov processes. I. Z. Wahrscheinlichkeitsth. 22, 205–225.CrossRefGoogle Scholar
Lukacs, E. and Szász, O. (1952). On analytic characteristic functions. Pacific J. Math. 2, 615–625.Google Scholar
Olver, F. W. J. (1974). Asymptotics and Special Functions. Academic Press, New York.Google Scholar
Pantí, H. (2012). On Lévy processes conditioned to avoid zero. Preprint. Available at http://arxiv.org/abs/1304.3191v1.Google Scholar
Pardo, J. C. (2009). The upper envelope of positive self-similar Markov processes. J. Theoret. Prob. 22, 514–542.Google Scholar
Rivero, V. (2007). Recurrent extensions of self-similar Markov processes and Cramér's condition. {II}. Bernoulli 13, 1053–1070.Google Scholar
Rogozin, B. A. (1972). The distribution of the first hit for stable and asymptotically stable walks on an interval. Theory Prob. Appl. 17, 332–338.Google Scholar
Song, R. and Vondra{ček, Z. (2006). Potential theory of special subordinators and subordinate killed stable processes. J. Theoret. Prob. 19, 817–847.CrossRefGoogle Scholar
Vigon, V. (2002). Simplifiez vos Lévy en titillant la factorisation de Wiener-Hopf. Doctoral Thesis, INSA de Rouen.Google Scholar
Vuolle-Apiala, J. (1994). Ito excursion theory for self-similar Markov processes. Ann. Prob. 22, 546–565.Google Scholar
Yano, K., Yano, Y., and Yor, M. (2009). On the laws of first hitting times of points for one-dimensional symmetric stable Lévy processes. In Séminaire de Probabilités XLII (Lecture Notes Math. 1979), Springer, Berlin, pp. 187–227.CrossRefGoogle Scholar